Optimal Bounds for Convergence of Expected Spectral Distributions to the Semi-Circular Law

TL;DR

This paper establishes optimal bounds on the rate at which the expected spectral distribution of certain Hermitian random matrices converges to the semi-circular law, showing a convergence rate of order O(n^{-1}).

Contribution

It provides the first optimal bounds for the convergence rate of expected spectral distributions to the semi-circular law for Wigner matrices.

Findings

Kolmogorov distance between distributions is O(n^{-1})

Convergence rate is proven under bounded fourth moments and growth conditions

Recursion method used to derive bounds

Abstract

Let $\mathbf X=(X_{jk})_{j,k=1}^n$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k\le n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf X$ to the semi-circular law assuming that ${\mathbf E} X_{jk}=0$, ${\mathbf E} X_{jk}^2=1$ and that $$ \sup_{n\ge1}\sup_{1\le j,k\le n}{\mathbf E}|X_{jk}|^4=:\mu_4<\infty \quad \text{and} \sup_{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac14}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$ and the semicircular law is of order $O(n^{-1})$.